Since their introduction in the 50's, variance component mixed models have been widely used in many application fields. In this context, ReML estimation is by far the most popular procedure to infer the variance components of the model. Although many implementations of the ReML procedure are readily available, there is still need for computational improvements due to the ever-increasing size of the datasets to be handled, and to the complexity of the models to be adjusted. In this paper, we present a Min-Max (MM) algorithm for ReML inference and combine it with several speed-up procedures. The ReML MM algorithm we present is compared to 5 state-of-the-art publicly available algorithms used in statistical genetics. The computational performance of the different algorithms are evaluated on several datasets representing different plant breeding experimental designs. The MM algorithm ranks among the top 2 methods in almost all settings and is more versatile than many of its competitors. The MM algorithm is a promising alternative to the classical AI-ReML algorithm in the context of variance component mixed models. It is available in the MM4LMM R-package.